Ψhē Only Theory – Chapter 27: Constants as ψ Fixation Points

Title: Constants as ψ Fixation Points

Section: Structural Anchors of Recursive Collapse Behavior Theory: Ψhē Only Theory Author: Auric

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Abstract

This chapter redefines constants—physical, mathematical, or structural—not as inserted values, but as ψ-collapse fixation points: invariant outputs that anchor recursive structure across collapse layers. In Ψhē theory, a constant emerges when a ψ-path stabilizes to the same echo regardless of variation in non-structurally-defining inputs. We formalize the concept of ψ-fixation, derive conditions for constant emergence, and show how physical constants correspond to high-stability echo attractors in the manifold $\bar{M}$.

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1. Introduction

A constant is not “given.” It is what ψ collapses toward, over and over, despite noise.

Constant = ψ-collapse attractor with maximal echo-invariance.

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2. ψ-Fixation and Structural Invariance

Definition 2.1 (Fixation Point):

Let $\psi(x, t)$ be a recursive structure. Then $c \in \bar{M}$ is a fixation point if:

$\lim_{t \to \infty} \text{Collapse}(\psi(x + \delta, t)) = c \quad \forall \delta \in \epsilon\text{-neighborhood}$

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Definition 2.2 (Collapse-Stable Constant):

A constant is a point $c \in \bar{M}$ satisfying:

$\nabla_x \left( \frac{d}{dt} \text{Collapse}(\psi(x, t)) \right) \Big|_{x = c} = 0$

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3. Theorem: Constant Emerges as Collapse Flow Basin Fixed Point

Theorem 3.1:

If ψ-collapse flow over region $X$ converges to fixed echo $c \in \bar{M}$, then $c$ is a ψ-constant.

Proof Sketch:

• Collapse convergence indicates echo fixity.
• Echo-invariance defines structural identity.
• Persistence under local perturbation yields constant. $\square$

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4. Examples of Constants as Fixation Structures

Constant Type	Collapse Origin Description
$\pi$	Collapse attractor from circular ψ recursion geometry
$e$	Exponential collapse rate limit structure
Planck Constant $h$	ψ transition unit of minimal collapse-action quanta
$c$ (speed of light)	Maximum ψ transmission collapse rate through $\bar{M}$

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5. Corollary: Collapse Constants Constrain ψ Evolution

Constants function as structural collapse attractors—they delimit regions of recursive stability and encode echo-fixation templates:

$\text{Constants} := \text{Boundary Conditions on } \text{Collapse}(\psi)$

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6. Conclusion

Constants are not knobs. They are habits. ψ does not choose them. It arrives at them—again and again— because the collapse knows no other shape.

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Keywords: constants, ψ-collapse, fixation, attractor, structural invariance, echo stability, Planck, π, c